Semi-feasible Algorithms
Hem-Ogi Chapter 3
CSC 286/486
Fall 2004
University of Rochester
Anustup Choudhury, Ding Liu, Eric Hughes, Matt Post, Mike
Spear, Piotr Faliszewski
Note for digital viewers


This presentation was put together using
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Semi-feasible Algorithms
Hem-Ogi Chapter 3
Section 3.1
Brought to you by:
Anustup Choudhury, Ding Liu, Eric Hughes,
Matt Post, Mike Spear, Piotr Faliszewski
Outline


Introduction
Definitions



Tools



P-sel
P/poly
Tournaments
Superloser Theorem
Main Result


P-sel  P/poly
P-sel  P/quadratic
Semi-feasible Problems




P – the class of feasible problems
A nice class, but lacks many interesting languages...perhaps
semi-feasible algorithms are also useful?
Let us consider languages that may not have polynomial-time
algorithms, but for which it is possible to efficiently decide which
of the two strings given is more likely to be in the language
One such decision does not give a definite answer as to
whether a chosen string is in a language, but perhaps a series
of them leads to a solution (of some nice problem)
P-sel • Definition

Def.
Set B is P-selective if there exists a function f,
f: S* x S* → S*
such that:



f is polynomial-time computable
f(x,y) ∈ {x,y}
{x,y} ∩ B ≠ ∅ ⇒ f(x,y) ∈ B
In other words: f always picks one of its inputs, and if
it can pick a one that is in B then it does so.
“Function f picks the input that is no less likely to
be in the set.”
Such a function f is called a selector function
P-sel • Definition (cont’d)

Def.


P-sel is the set of of all P-selective languages
P ⊆ P-sel


The selector function can simply test membership
of both of its arguments
Let A ∈ P. We define the selector function f to be:



f(x,y) = x if x ∈ A
f(x,y) = y otherwise
Is P a proper subset of P-sel?


Yes! That is exactly your homework!
Don’t worry, we will help 
P-sel • Example


So what sets might be in P-sel, but outside of P?
Consider the language:
LW = { x | x ≥ W }
Where:

W is some real number

the second occurrence of x is treated as a real number whose
binary representation is x

Why is this language P-selective?


f(x,y) = max(x,y)
What is so special about it?
Recall the Halting Problem...

Def.
HP = { x | x ∈ L(Mx) }


As we all remember, HP is in RE–RECURSIVE
Can we use that to our advantage?
P-sel • Example (Cont’d)



Gregory Chaitin from IBM found a really nice W…
Note: the first occurence of p is treated as a string, but
the second as an integer
If
were recursive then RECURSIVE=RE

But you still need to prove that!
P-sel • NP-hard sets in P-sel?


P-sel contains sets that are not even
decidable!
It stands to reason that there is some NPcomplete language that is P-selective!


Um... Er... Well not likely, actually...
Theorem
If there exists an NP-hard language A such that A is
P-selective then P=NP
The opposite direction is clearly true. We have P ⊆ P-sel so
If P=NP then all NP-complete sets are in P-selective. NPcomplete sets are NP-hard.
P-sel • Proof of the previous theorem
Assumptions:




A – an NP-hard
language
A ∈ P-sel
g – a selector function
for A
f – a polynomial-time
computable function
many-one reducing
SAT to A
Let F be the input formula. How do we use f
and g to find a satisfying truth assignment?
F
F[v=True]
F[v=False]
g( f( F[v=True]) , f( F[v=False] ) )
g picks the best path!
Proof technique:
g is a selector function—it has to return the
string that is no less likely to be in the set.
Show a polynomial-time
algorithm for SAT by
pruning the tree of
possible truth assignments
Since f is a reduction, g gives us the path in
the tree to follow! After n steps the ground
formula is guaranteed to be satisfiable iff F
is satisfiable.
P-sel • Conclusion



A set is P-selective if given two strings we can decide
in polynomial time which of them is no less likely to
be in the set
P-selective sets may be undecidable
Yet, unless P=NP, none of them can be NP-hard
P/poly • Introduction

Recall tally languages


L is tally if L ⊆ 1*
What does it take to decide a tally language?



L could be undecidable!
To decide whether a string of length n is in a tally
language you just need to know whether 1n is in
the language...
With 1 bit of advice for every length we could
decide any tally language...
P/poly • Introduction (Cont’d)

What about sparse languages?

L is sparse if there is a polynomial p such that:


|L=n| ≤ p(n)
How much advice per length do we need to decide sparse
languages?




There are at most polynomially many strings of length n
Each string is n symbols long
If, as a piece of advice, someone gave us all the strings of the
given length then we could simply compare each with the input
The advice would only be polynomial in size
P/poly • Definition (Informal)




P/poly is the class of all languages that can
be decided given a polynomial amount of
advice
A P/poly advice interpreter has to work in
polynomal time, but for every string length it is
given polynomially many advice bits
Advice is anything the P/poly algorithm
designer wants; it does not even have to be
computable…
...but it must only depend on the input length
and not on the input content
P/poly • P/poly and sparse sets

Theorem
All tally sets are in P/poly

Theorem
All sparse languages are in P/poly

In fact, P/poly is exactly the class of all sets
Turing-reducible to sparse sets.

Unfortunately, we do not have time to prove this
P/poly • Definition of A/f

What do the symbols mean?



A/f is the class of all languages L such that for some
function h it holds that:



A – some language
f: N → N – some function
(∀n)[ |h(n)| = f(n) ]
L = { x | <x, h(|x|)> ∈ A }
Intuitions:



Function f measures the amount of advice available
Language A is the advice interpreter
Function h provides the advice
P/poly • Definition of C/F

What do the symbols mean?



C/F is the class of all languages L such that :


C – a class of languages
F – a set of functions from integers to integers
(∃ A ∈ C) (∃ f ∈ F) [ L ∈ A/f ]
Where is P/poly?

Take C to be the class P, and make F the set of all
polynomials
P/poly • Example



Let us formally prove that all tally languages are in P/poly
Let T be some tally language
First, select the advice function:

We only need 1 bit of advice; f(n) = 1



Advice interpreter:


h(n) = 1 if 1n ∈ T
h(n) = 0 otherwise
A = {<x,y> | x ∈ 1* and y=1}
A ∈ P, f is a polynomial ⇒ T ∈ P/poly
Outline

Piotr defined for us:



P-sel
P/poly
Here’s what I’m going to do


Explain k-tournaments
Prove that P-sel ⊆ P/poly
Thm 3.1 • k-tournaments

What is a k-tournament?


a graph with k nodes, and exactly one directed
edge between every pair of vertices
Why is it called a “tournament”?

think of each edge as a game played, with the
arrow pointing at the winner
Our k-tournament (k=8)
2
3
1
4
8
5
7
6
• arrows point to
the winners
• each node is
also considered
to defeat itself
Thm 3.1 • the superloser set
Properties of a k-tournament


G = a k-tournament graph
H⊆G
1)
2)


for each v ∈ VG–H, there is some g ∈ H such
that (g,v) ∈ EG
in other words, there is a subset H of G whose
cardinality is O(log n) in the number of nodes in
G, and every node in G defeats one of the nodes
in H
we call H the superloser set
Proof 3.1 • process


At least one person loses half or more of her games
(why?)
Proof procedure: Take that player and remove her
from the graph, G, as well as everyone who defeated
her. Add the player to the loser set, H


there are between 0 and
nodes left
Repeat this process on the remaining graph, G’, until
there are no nodes left.
A Visualization of Thm 3.1
2
3
1
4
8
5
H = {8}
{8,6}
7
6
find the
allfind
remove
nodes
every
select
complete
thewe’ll
losers
that
node
all
allarbitrarily
nodes
graph
of
we’ll
lost
defeats
on
them,
half
choose
the
that
with
or
add
choose
remaining
abeat
the
more
super-loser
6
the
superlosers
our
8
loser
of loser
graph
their
to games
H
Proof of Thm 3.1 (cont’d)



At each stage we have a full k-tournament
graph, with k shrinking through successive
stages
Eventually there will be no nodes to consider
The recurrence relation for this is
Review so far

As Piotr explained:

the class P-sel…


the class P/poly


“semi-feasible” sets: the class of sets that have a selector
function that takes two arguments, and returns the one
more likely (not less likely) to be in the set
sets that can be solved with advice that is the same for
all strings of the same length
As I explained

k-tournaments
Hem-Ogi Thm 3.2 • P-sel ⊆ P/poly


Proof idea: turn a semi-feasible set into a
P/poly set using a k-tournament
Potential point of confusion: the text talks
about showing “that semi-feasible sets have
small circuits”

having small circuits is another characteristic of
P/poly; just ignore for now
Proof 3.2 • goal


Let L be our semi-feasible (P-sel) set
To show that it’s in P/poly, we need to show
two things
1.
–
A will be the advice interpreter set from the definition of
P/poly
2.
–
q is the polynomial bound on the advice size
Proof 3.2 • k-tournament at L=n


We’ll begin by making a k-tournament from
the elements of L=n.
How?



remember that a k-tournament is a property of any
fully-connected directed graph
we can consider each string in L=n to be a node,
so all we need is a way to decide which of two
strings is the “winner”
Any ideas?
Proof 3.2 • selector function


L is a semi-feasible set, so there is a selector
function f
Let f’(x,y) = f(min(x,y), max(x,y))



f’ is a selector function for the same language
f’ is commutative
The commutativity of f’ allows us to construct
a k-tournament from the strings in L=n

call this graph G such that for any
(a,b ∈ L=n ∧ a ≠ b) ⇒ ((a,b) ∈ EG ⇔ f’(a,b) = b)
Proof 3.2 • Properties of L=n

Because of the k-tournament, we know that
we have a superloser set Hn ⊆ L=n where:
1)
for every element in L=n, there exists an
h ∈ Hn such that f(h,x) = x
(remember that every superloser beats itself)
2)
Proof 3.2 • map of the world
a map
of the world
Proof 3.2 • what advice?

So far we have used the selector function to
produce a k-tournament: Now we’ll show our
set L is in P/poly by providing




an advice function, g, and
an advice interpreter, the set A
Remember that advice on a set is the same
for all strings of the same length
Any guesses what advice g gives for L on
strings of length n?
Proof 3.2 • good advice



The advice is Hn: g(n) provides the
superloser set at length n
The advice interpreter set is:
A = {<x,y> | y is a (possibly empty)
list of elements v1, v2, … ,vz and for
some j it holds that f(vj,x) = x.
Clearly, A ∈ P. Proof:
On input <x,y>
FOR j FROM 1 TO z DO
IF f(x,vj) == x ACCEPT
REJECT
Proof 3.2 • correctness

Are the requirements met?


A∈P
on the next slide we’ll show that x is in L if and
only if <x,g(|x|)> is in the advice interpreter set A
(i.e., that we’ve met the requirement for a set in
P/poly)
Proof 3.2 • verification

Verification of MA: three cases


x is a superloser, so the test for some hj is whether f(x,x)
= x, which is always true


x is not a superloser, but since it is in L it will defeat one
of the superlosers, so we accept


x is not a superloser and does not defeat one, so we
reject
One for the road

The class P/poly allows a polynomial number
of advice bits. How many did we just use,
and what does that do for us?
Semi-feasible Algorithms
Hem-Ogi Chapter 3
Section 3.2
Brought to you by:
Anustup Choudhury, Ding Liu, Eric Hughes,
Matt Post, Mike Spear, Piotr Faliszewski
Outline

Review




Limited Advice



P-sel, P/poly
P-sel ⊆ P/poly
k-Tournaments
k-Tournament properties
P-sel ⊆ NP/linear
Setting the stage for a Polynomial Hierarchy
collapse
P-sel

Set B is P-selective iff ∃ function f




f is polynomial-time computable
f(x,y) ∈ {x,y}
{x,y} ∩ B ≠ ∅ ⇒ f(x,y) ∈B
B has a polynomial-time 2-ary selector
function which always chooses the input
more likely to be in B
P/poly

The class of all languages that can be
decided in polynomial time with a polynomial
amount of advice



The advice is polynomial with regard to the length
of the string whose membership is being tested
Advice is dependent only on input length, not input
content
The advice doesn’t even have to be
computable
k-Tournaments

A graph with k nodes, and exactly one
directed edge between every pair of vertices



No self loops
Each arrow represents a game played, with
an arrow pointing to the winner
There is a superloser set of size ≤ ⎣log(k+1)⎦
P-sel ⊆ P/poly


If we treat the members of our P-sel language
as a tournament, then we can use the
superloser set as advice
We can check the membership of an arbitrary
string x by applying the selector function on
every pair (x,y), where y is a string in the
superloser set
Limited Advice

Did we really only show P-sel ⊆ P/poly?


We only used a quadratic amount of advice!
Can we do better (perhaps by using some
nondeterminism?)
The Class PP


It can be shown that P-sel ⊆ PP/linear
But PP ⊇ NP
The Class NP



P/poly ⊆ NP/poly
P/quadratic ⊆ NP/quadratic
Can nondeterminism reduce the advice
needed to determine membership in
polynomial time?
k-Tournaments
Background
 The “l” nodes are a
superloser set
 Every node in column
x defeats node lx
 There are ⎣log(k+1)⎦
superlosers
 l1 beat l2 (otherwise, l2
would be in l1’s column)
 Likewise, l2 beat l3, l3
beat l4 …
Tournament
a1
a2
a3
a4
…
l1
b1
b2
b3
b4
…
l2
c1
c2
c3
c4
…
l3
d1
d2
d3
d4
…
l4
e1
e2
e3
e4
…
l5
f1
f2
f3
f4
…
l6
g1
g2
g3
g4
…
l7
h1
h2
h3
h4
…
l8
How much advice is
needed?
●
No more than the number
of superlosers
●
When x ∈ L, x defeats at
least one superloser
●
When x ∉ L, x does not
defeat any superloser
The King Loser





Look at l8 more closely
Every superloser beat l8
 If not, l8 would be in
another loser’s
column
Everyone else beat a
superloser
Everyone in the
tournament is “2 hops”
from l8
l8 is the KING LOSER
a1
a2
a3
a4
…
l1
b1
b2
b3
b4
…
l2
c1
c2
c3
c4
…
l3
d1
d2
d3
d4
…
l4
e1
e2
e3
e4
…
l5
f1
f2
f3
f4
…
l6
g1
g2
g3
g4
…
l7
h1
h2
h3
h4
…
l8
The King Loser Theorem


If G is a k-tournament, then there is a v ∈ VG
such that VG = R2,G(v)
In every k-tournament, there exists a node
from which all nodes can be reached via
paths of length 2 or less
Proof of the King Loser Theorem





Proof by induction
Base case: for k-tournaments whose size ≤ 3, it is
obviously true
k=1
k=2
k=3
a
a
a
a
b
b
b
c
c
Node a is always a King Loser
Some graphs have several King Losers
Disclaimer: The k=1 base case would suffice for
the following proof
Proof of the King Loser Theorem


When k > 3, we use induction
Recall when k=3
a
b
c

For any k-tournament, we can classify every node
as follows:
1.
2.
3.
x is the king loser
x beat the king loser
x beat someone who beat the king loser


(this implies that the king loser beat x)!
This classification into sets is clear in our k=3
example above

(a is in set 1, b is in set 2, c is in set 3)
Proof of the King Loser Theorem

What happens when we add a new node?

Case 1: The new node (d)
beats the king loser (a)
a
d
B
C


Case 2: The new node (d)
beats someone who
beats the king loser
(that is, some node in set B)
a
d
B
C
(B and C are sets and may
contain multiple nodes
Either way, the King Loser doesn’t change
Proof of the King Loser Theorem

Case 3: If cases 1 and 2 do not hold, then
the new node becomes the king loser

The king loser (a) beat the new node
(d)


Everyone who beat the king loser
(everyone in set B) beat the new
node (d)


otherwise case 1
otherwise case 2
Everyone else (all nodes in set C) is
no farther from the new node (d)
than from the old superloser (a)!

c beat b, b beat d, and b beat a
a
d
B
C
(B and C are sets and may
contain multiple nodes
Using the King Loser


Let A ∈ P-sel via commutative selector function f.
Consider using f to build a tournament on the nodes in A=n
(we’ll be vague on uniformity but it isn’t a problem here)
If we knew the King Loser, we could use the following
algorithm to determine the membership of x in A=n:
If x = King Loser, accept
ElseIf f(x, King Loser)=x, accept
Else
Nondeterministically guess a string
y of the same length as x
On each path, if f(x,y)=x and
f(y,King Loser)=y, accept
Reject
Encoding the King Loser



We want to show that P-sel ⊆ NP/linear
The King Loser looks like sufficient advice
How do we encode the King Loser?


There is a distinct King Loser for each length n
What if L=n = ∅ ?
Using n+1 Bits

Let us define the advice function g(x) as
follows:

where wn is the King Loser for strings of
length n
Using n+1 Bits



Let us define the advice interpreter
A = {〈x, 0w〉⎟there is a path of length at most
two, in the tournament induced on L=n by f,
from w to x}
We hard-code the case of ε ∈ L
Since the selector function f is deterministic
and takes polynomial time, we can construct
a NPTM to decide A.
Using n+1 Bits

Recall this algorithm from before
If x = King Loser, accept
ElseIf f(x, King Loser)=x, accept
Else
Nondeterministically guess a string
y of the same length as x
On each path, if f(x,y)=x and
f(y,King Loser)=y, accept
Reject

Since f is a deterministic polynomial-time function,
this is clearly a nondeterministic polynomial-time
algorithm
Conclusions


P-sel ⊆ NP/linear
Since P-sel is closed under complementation




P-sel ⊆ coNP/linear
P-sel ⊆ NP/linear ∩ coNP/linear
P-sel ⊆ NP/n+1
Can we do better?

NO (see the book for details)
Collapsing the Polynomial Hierarchy


Using the techniques we’ve covered so far,
we can learn the more subtle properties of
the polynomial hierarchy
But what is the polynomial hierarchy?
The Polynomial Hierarchy


A time-bounded analog of the arithmetical
hierarchy
We can think of it iteratively…



Q, R are poly-time predicates; p’, p’’ are polynomials
swaps the
quantifiers
… or inductively
The Polynomial Hierarchy
(bounded by PSPACE)
FP – Deterministic Functions

A function is in FP iff:



It is single valued
It is computed by a deterministic, polynomial time
TM
It does not have to be total
Nondeterministic Functions


NPMV – nondeterministic, polynomial time,
multi-valued
A function f belongs to NPMV if there exists a
NPTM N such that on input x, f’s outputs are
exactly the outputs of N
y1 y2 y3 y2 y4 y2
y5
Nondeterministic Functions

On input x, set-f(x) is the set of all possible
outputs of NPMV function f



set-f(x) = {a⎟a is an output of f(x)}
On inputs where f(x) is undefined, set-f(x) = ∅
We don’t care if an item in set-f(x) occurs on
multiple paths
set-f(x) = {y1, y2, y3, y4, y5}
y1 y2 y3 y2 y4 y2
y5
Nondeterministic Functions

NPSV – nondeterministic, polynomial time,
single-valued

A subset of NPMV where ∀ x, ||set-f(x)|| ≤ 1
set-f(x) = {y1}
y1 y1 y1 y1 y1 y1
y1
NPMV Selector Functions

A set L is NPMV-selective if





∀ x,y. set-f(x,y) ⊆ {x,y}
∀ x,y. x ∈ L ∨ y ∈ L ⇒ ∅ ≠ set-f(x,y) ⊆ L
In other words, NPMV-selector functions can
return multiple values, but only if both
arguments are in L or both arguments are not
in L
The selector function can return ∅ when both
arguments are not in L
NPSV-selective sets exist as well
Refinement

NPMV function f is a refinement of NPMV
function g if




∀ x. set-f(x) = ∅ ⇔ set-g(x) = ∅
∀ x. set-f(x) ⊆ set-g(x)
A refinement has fewer outputs, but remains
defined whenever the original function was
defined
A refinement may be NPSV
Next Time

Section 3.3: Unique Solutions Collapse the
Polynomial Hierarchy
Semi-feasible Algorithms
Hem-Ogi Chapter 3
Section 3.3
Brought to you by:
Anustup Choudhury, Ding Liu, Eric Hughes,
Matt Post, Mike Spear, Piotr Faliszewski
Outline

Review





FP, NPMV, NPSV
refinements
NPMV-sel, NPSV-sel
NP/poly, coNP/poly etc.
Goal


If all NPMV functions have NPSV refinements then
PH collapses to its second level...
... and even further
FP – Deterministic Functions

A function is in FP iff:



It is single-valued
It is computed by a deterministic, polynomial-time
TM
It does not have to be total
Nondeterministic Functions


NPMV – nondeterministic, polynomial-time,
multivalued
A function f belongs to NPMV if there exists a
NPTM N such that on input x, f’s outputs are
exactly the outputs of N
y1 y2 y3 y2 y4 y2
y5
Nondeterministic Functions

On input x, set-f(x) is the set of all outputs of
NPMV function f



set-f(x) = {a⎮a is an output of f(x)}
On inputs where f(x) is undefined, set-f(x) = ∅
We don’t care if an item in set-f(x) occurs on
multiple paths
set-f(x) = {y1, y2, y3, y4, y5}
y1 y2 y3 y2 y4 y2
y5
Nondeterministic Functions

NPSV – nondeterministic, polynomial-time,
single-valued

A subset of NPMV where ∀ x, ||set-f(x)|| ≤ 1
set-f(x) = {y1}
y1 y1 y1 y1 y1 y1
y1
NPMV Selector Functions

A set L is NPMV-selective if





∀ x,y. set-f(x,y) ⊆ {x,y}
∀ x,y (x ∈ L ∨ y ∈ L) ⇒ ∅ ≠ set-f(x,y) ⊆ L
In other words, NPMV-selector functions can
return multiple values, but only if both
arguments are in L or both arguments are not
in L
The selector function can return ∅ when both
arguments are not in L
NPSV-selective sets exist as well
Refinement

NPMV function f is a refinement of NPMV
function g if




∀x. set-f(x) = ∅ ⇔ set-g(x) = ∅
∀x. set-f(x) ⊆ set-g(x)
A refinement has fewer outputs, but remains
defined whenever the original function was
defined
A refinement may be NPSV
Goal

Theorem
If all NPMV functions have NPSV refinements then PH
collapses to its second level, NPNP.

We need the following intermediate results:



NPSV-sel ∩ NP ⊆ (NP ∩ coNP)/poly
NP ⊆ (NP ∩ coNP)/poly ⇒ PH=NPNP
Proof outline




Create an NPMV selector for SAT
Refine it to be an NPSV selector
Conclude NP ⊆ NPSV-sel ∩ NP
NP ⊆ NPSV-sel ∩ NP ⊆ (NP ∩ coNP)/poly ⇒ PH = NPNP
Goal

Theorem
If all NPMV functions have NPSV refinements then PH
collapses to its second level, NPNP.

We need the following intermediate results:



NPSV-sel ∩ NP ⊆ (NP ∩ coNP)/poly
NP ⊆ (NP ∩ coNP)/poly ⇒ PH=NPNP
Proof outline




Create an NPMV selector for SAT
Refine it to be an NPSV selector
Conclude NP ⊆ NPSV-sel ∩ NP
NP ⊆ NPSV-sel ∩ NP ⊆ (NP ∩ coNP)/poly ⇒ PH = NPNP
Goal

Theorem
If all NPMV functions have NPSV refinements then PH
collapses to its second level, NPNP.

We need the following intermediate results:



NPSV-sel ∩ NP ⊆ (NP ∩ coNP)/poly
NP ⊆ (NP ∩ coNP)/poly ⇒ PH=NPNP
Proof outline




Create an NPMV selector for SAT
Refine it to be an NPSV selector
Conclude NP ⊆ NPSV-sel ∩ NP
NP ⊆ NPSV-sel ∩ NP ⊆ (NP ∩ coNP)/poly ⇒ PH = NPNP
Roadmap






Assume all NPMV functions have NPSV
refinements
Prove: SAT has an NPMV selector function
If:
SAT has an NPSV selector function
Then: NP ⊆ NP ∩ NPSV-sel
Prove: NP ∩ NPSV-sel ⊆ (NP ∩ coNP)/poly
Prove: NP ⊆ (NP ∩ coNP)/poly ⇒ PH=NPNP
NPMV selector for SAT

NPMV selector for SAT:



fSAT(x,y) = {x,y} ∩ SAT
Clearly, fSAT is a selector
Is it in NPMV?




Nondeterministically choose x or y
Guess a satisfying truth assignment for the string chosen
Check if it indeed is satisfying, and output the chosen
string if so
SAT ∈ NPMV-sel
NPSV selector for SAT

Assumptions


All NPMV functions have NPSV refinements
fSAT is an NPMV selector for SAT

We can refine it to be an NPSV selector!
Roadmap






Assume all NPMV functions have NPSV
refinements
Prove: SAT has an NPMV selector function
If:
SAT has an NPSV selector function
Then: NP ⊆ NP ∩ NPSV-sel
Prove: NP ∩ NPSV-sel ⊆ (NP ∩ coNP)/poly
Prove: NP ⊆ (NP ∩ coNP)/poly ⇒ PH=NPNP
NP ⊆ NP ∩ NPSV-sel

Assumptions



All NPMV functions have NPSV refinements
Under these assumptions SAT is in NPSV-sel
(it has an NPSV selector function)
If NPSV-sel was closed under polynomialtime many-one reductions then we would be
done
NPSV-sel


Theorem
NPSV-sel is closed under many-one
polynomial-time reductions
Proof




A polynomial-time many-one reduces to B
B ∈ NPSV-sel
fB – NPSV selector for B
g – FP function many-one reducing A to B
Roadmap






Assume all NPMV functions have NPSV
refinements
Prove: SAT has an NPMV selector function
If:
SAT has an NPSV selector function
Then: NP ⊆ NP ∩ NPSV-sel
Prove: NP ∩ NPSV-sel ⊆ (NP ∩ coNP)/poly
Prove: NP ⊆ (NP ∩ coNP)/poly ⇒ PH=NPNP
NPSV-sel ∩ NP ⊆ (NP ∩ coNP)/poly

Assumptions





L ∈ NPSV-sel ∩ NP
NL – an NPTM accepting L
f – an NPSV selector for L
set-f(x,y) = set-f(y,x)
Goal



Show that L ∈ (NP ∩ coNP)/poly
Proof is essentially the same as in section 3.1, but
with a more carefully chosen advice string.
We need to provide:


An advice interpreter that belongs to NP ∩ coNP
Advice of at most polynomial size
NPSV-sel ∩ NP ⊆ (NP ∩ coNP)/poly

Advice interpreter:


Input: <x,d>
Interpret d as <<a1,…,am>,<w1,…,wm>>



Output:




Accept if for every i, wi is an accepting computation path of NL
on ai, and set-f(x,aj) = {x} for at least one j
Reject otherwise
Clearly, it is an NP algorithm
It is also a coNP algorithm


List of strings a1, …, am, each of length |x|
List of strings w1, …, wm
Discussed during lecture
The advice


ai – superloser set for a tournament induced by f on L=n
wi – accepting computation paths for ai’s.
Roadmap






Assume all NPMV functions have NPSV
refinements
Prove: SAT has an NPMV selector function
If:
SAT has an NPSV selector function
Then: NP ⊆ NP ∩ NPSV-sel
Prove: NP ∩ NPSV-sel ⊆ (NP ∩ coNP)/poly
Prove: NP ⊆ (NP ∩ coNP)/poly ⇒ PH=NPNP
Relativization

Theorem X
If A ∈ P/poly then Ā ∈ P/poly

Proof



Advice interpreter for Ā simulates the advice
interpreter for A, and flips its answer
Advice is the same
Relativized version Theorem X
If A ∈ PB/poly then Ā ∈ PB/poly
 The same proof works!
 We say that Theorem X relativizes
Relativization (Cont’d)

Most of theorems relevant to complexity
theory relativize



There are some exceptions, though.
Nonrelativizing theorems are usually very hard to
prove
A theorem resolving the P vs. NP problem
cannot relativize


There is a set A such that PA = NPA
There is a set B such that PB ≠ NPB
NP ⊆ (NP ∩ coNP)/poly ⇒ PH=NPNP

The Karp-Lipton Theorem


NP ⊆ P/poly ⇒ PH=NPNP
This theorem relativizes



Assumptions



Let A be some language
A
NPA ⊆ PA/poly ⇒ PHA=NPNP
NP ⊆ (NP ∩ coNP)/poly
SAT in (NP ∩ coNP)/poly via NP ∩ coNP set B
Proof




B
NPB ⊆ PB/poly ⇒ PHB = NPNP
NPB = NP because NPNP ∩ coNP = NP, PHB = PH
NP ⊆ PB/poly ⇒ PH=NPNP
NP is a subset of PB/poly because


SAT in PB/poly
PB/poly closed under many-one reductions
Conclusion

We have reached our goal!



We proved all intermediate results
It holds that if all NPMV functions have an NPSV refinement
then PH collapses to its second level
Interpretation



What does it mean for an NPMV function to have an NPSV
refinement?
It means that an NPTM can isolate a single solution from
possibly exponentially many
Isolating a unique solution for every NPMV function
collapses the polynomial hierarchy to its second level